Let $\displaystyle f:R\to R$ be a twice differentiable function and suppose that for all $\displaystyle x\in R$,$\displaystyle f$ satisfies the following two conditions:

(i)$\displaystyle |f(x)|\leq 1$

(ii)$\displaystyle |f''(x)|\leq 1$

Prove that $\displaystyle |f'(x)|\leq 2$